where the local wall streamwise velocity gradient is
η0≡ ∂eu
∂z 0
. (4.10)
The local wall shear stress isτ0(x, y, t) =νη0(x, y, t)≡u2τ(x, y, t) whereuτ(x, y, t) is the local wall- friction velocity. Since ∆f l+, each computational cell is effectively assumed to behave as a local statistically homogeneous unit of wall turbulence, similar to the so-called minimal flow unit of Jim´enez and Moin (1991). Experiments based on one-dimensional filtering conducted by Nakayama, Noda and Maeda (2004) suggest ∆f >1800l+. When two-dimensional filtering is employed, as in the present LES application, they argued that filter widths larger than ∆+x ×∆+y = 200×360 are adequate.
4.3.2 Local–Inner Scaling
We now introduce a local–inner-scaling ansatz. This states that the statistics of each cell are char- acterized by their respective local–inner scales ν and η0(x, y, t) or equivalently l+ and uτ(x, y, t).
Specifically, we assert that the SGS streamwise velocity, within a near-wall, sugbrid region to be defined subsequently, can, for each cell, be collapsed onto the form
u(x, y, z, t) = (νηe 0(x, y, t))1/2F(z+), z+=z(η0(x, y, t)/ν)1/2≡z/l+, (4.11) whereF(z+) can be thought of as a local “law of the wall”. Next, differentiate (4.11) with respect toη0to give
∂ue
∂η0 = 1
2(ν/η0)1/2
F+z+F0
, (4.12)
where F0 = dF/dz+; perform the wall-adjacent average (4.8); and then back-substitute the inner scaling (4.11), evaluated atz=h,
∂hui
∂η0
=1
2(ν/η0)1/2F(h+) = u|eh 2η0
,
where u|eh=F(h+) withh+=h/l+. Finally, using the chain rule we find
∂hui
∂t = ∂hui
∂η0
∂η0
∂t = u|eh 2η0
∂η0
∂t . (4.13)
We emphasize that (4.13) is an exact consequence of (4.8) and (4.11). Moreover, using (4.13) to evaluate the time derivative of the plane-filtered and vertically averaged streamwise velocity, the explicit form of F(z+) in 0 ≤ z < h is not needed; this occurs owing to the cancellation of two integrals. The velocity eu|hwill later be obtained directly from the LES.
Now substitute (4.13) into the first term of (4.9) to obtain u|eh
2η0
∂η0
∂t +∂huui
∂x +∂huvi
∂y =−1
h uw|f h− ∂pe
∂x h
+ν h
∂ue
∂z h
−η0
. (4.14)
Our motivation for performing the wall-adjacent average, (4.8), is to remove the steep near-wall gradients, which we do not wish to resolve. In-plane (x, y) gradients of filtered quantities in (4.14) are now approximated by values atz=h, supplied by the LES,
∂huui
∂x ≈∂uu|fh
∂x , ∂huvi
∂y ≈ ∂uv|fh
∂y .
This approximation captures some of the nonequilibrium effects arising from large-scale in-plane inhomogeneities. With these assumptions, we rewrite (4.14) as
∂η0
∂t = 2η0 u|eh
−1
h uw|f h−∂uu|fh
∂x −∂uv|fh
∂y − ∂pe
∂x h
+ν h
∂eu
∂z h
−η0
. (4.15)
Equation (4.15) governs the evolution of the wall shear stress, written in terms ofη0. Viewed as a standalone entity, we treat (4.15) as an ODE driven by external forcing, even though it is strictly a partial differential equation when coupled with the LES. The right-hand side of (4.15) only involves known quantities ath, which is arbitrary. In practice, we choosehto be at the first grid point within the LES (see figure 4.1), and the quantities on the right-hand side can then be determined from resolved-scale LES quantities. We remark that our use of local–inner scaling, (4.11), is restricted to the reduction of the unsteady term in (4.9) and that this operation does not require a specific form for F(z+). The other terms in (4.15) will be provided from the resolved-scale LES itself, so that (4.15) can then be viewed in this sense as resulting from mixed inner–outer scaling. When coupled with an LES, (4.15) then allows us to determine the wall shear stress without resolving the near-wall steep gradients, which have been integrated out by the wall averaging. One can also interpret (4.15) as an integrated form of the local unsteady turbulent boundary layer equations with the added assumption of local–inner scaling for the unsteady term. Further, (4.15) knows nothing about the channel geometry and should, therefore, be applicable to general flows. To close this coupling, appropriate boundary conditions for the LES need to be applied, which is the subject of the next subsection.
4.3.3 Multilayer SGS Wall Model
We do not resolve the near-wall region. Instead, the LES computation takes place above a certain fixed,Re-independent heighth0, which will later be chosen as a small fraction of the near-wall cell size. To proceed we first define three regions for the lower half channel. It is understood that, for the present simulations, similar regions exist on the upper wall. These regions are (see figure 4.1)
h0
h Δx
Δz
hν
(II) (III)
(I) u|hν
u|h0
ev =ex
ev =eSe
Figure 4.1. Schematic showing the near-wall setup: h0locates the lifted virtual wall, where boundary conditions are applied;hlocates the input plane to the wall shear stress equation, (4.14);hν locates the outer edge of the viscous sublayer;evis the alignment of SGS vortices in their respective regions.
(i) 0≤z≤hν, region (I), essentially the viscous sublayer;
(ii) hν < z ≤h0, region (II), viewed as an overlap layer, where the shear stress is approxi- mately constant and will be modeled by the extended stretched-vortex SGS model consisting of attached vortices aligned withex; and
(iii) h0< z≤δ, region (III), where nonuniversal outer flow features are computed with LES coupled with the original stretched-vortex SGS model of detached subgrid vortices aligned withe
Se.
We remark that the combination of attached and detached vortices was also used by Maruˇsi´c and Perry (1995) to model wall turbulence. The planez=hlies at the top of the first grid cell in region (III). The planez=h0 will be referred to as the lifted virtual wall. We now proceed to model the flow in regions (I) and (II) in a way that provides a slip velocity atz=h0.
In region (I) we use ue+ =z+, where ue+ =u/ue τ, z+ =z/l+, and uτ is known. In particular, ue+|hν =h+ν, whereh+ν =hν/l+. For a hydrodynamically smooth wall, where the wall roughness is small compared to l+, experiments indicate that the outer edge of the viscous sublayer is located at h+ν ≈11 (based on the intercept between the linear and log components of the law of the wall).
We will therefore takeue+|hν =h+ν = 11. In fact, this intercept is found to be sensitive to pressure gradients and can assume values in the range 10–15 (Nickels 2004). A cubic equation was successfully used in the paper of Nickels (2004) to model this effect; we do not pursue this presently in favor of simplicity, although this generalization should certainly be included for separating flows. Above z+=h+ν, inviscid outer flow dynamics become important.
4.3.4 Slip Velocity at Lifted Virtual Wall
We now model the mean-flow dynamics in region (II), hν < z ≤h0. We require h0 to scale with outer flow thickness δ but to remain relatively small, h0 <0.1δ, say, so that nonuniversal effects (the wake) uncharacteristic of the inner scales can be captured by the LES in region (III). This will permit the LES to be performed with the same grid for a wide range ofRe, eliminating theO(Re2τ) scaling requirement for the grid resolution of a partially resolved wall-bounded LES (Piomelli 2008).
Put another way,h0 remains fixed,O(δ), but hν becomes thinner,O(l+), with increasingRe.
Region (II) is, by construction, the so-called overlap region, or the production-equals-dissipation layer, where the shear stress is approximately constant. Furthermore, the shear stress is balanced by the wall shear stress (Townsend 1976). Casting these ideas in LES terminology,
u2τ(x, y, t) =−uwf =−uewe−Txz =−Txz, sincewe= 0.
The existence of quasi-streamwise vortical structures in wall turbulence have long been observed by researchers (e.g., Head and Bandyopadhyay 1981, Robinson 1991), and have also served as useful physical models (e.g., Bakewell and Lumley 1967, Townsend 1976, Perry and Chong 1982, Maruˇsi´c and Perry 1995, Nickels et al. 2007, Adrian 2007). Motivated by these studies, we model region (II) with an ensemble of vortices aligned in the streamwise direction, (evx, evy, evz) = (1,0,0)⇔ev =ex. Substituting these into the expression for the shear stress produced by the extended stretched-spiral vortex SGS model, (4.5), and noting that the only nonzero component of the mean velocity gradient tensor is du/dz, we obtaine
Txz=−1
2γIIK1/2∆c
deu
dz. (4.16)
Recall the physical mechanism that produces this shear stress: the action of the spiralling streamwise vortex is to wrap its own axial velocity, now identified as the mean streamwise velocity, as if it were a passive scalar (see figure 4.2), thereby transporting higher-momentum fluid toward the wall and transporting low-momentum fluid away from the wall. This process has the observed effect of a flattened streamwise velocity profile.
Unlike the SGS vortices in region (III), which are unaware of the presence of the wall and are, therefore, considered as detached from the wall, the size of these near-wall vortices, ∆c, are constrained by the presence of the wall so that ∆c = z. That is, (4.16) with ∆c → z can be interpreted as the shear stress produced by a hierarchy of longitudinal vortices that scale with the wall distance. This scaling assumption is, in fact, the idea of the so-called attached wall eddy (Nickels et al. 2007). We therefore write (4.16) in the form
due dz = 1
K1
uτ
z , (4.17)
x z y
ev
t=t0 t=t1 t=t2
ev
Figure 4.2. Schematic of a pair of attached counterrotating vortices. The winding effect of the vortices are shown on contour plots of the streamwise velocity along with the accompanying profiles.
Darker shades represent higher-momentum fluid. The various stages of mixing are characterized by the timest=t0< t1< t2.
where the dimensionless local quantity given by
K1(x, y, t) = γIIK1/2
2 (−Txz/uτ) (4.18)
resembles the K´arm´an constantκ. An implicit assumption in the derivation of (4.17) is that K is sensibly independent of z, even though ∆c decreases as the wall is approached. This is possible if the number of these vortices (population density) also increase in proportion to their decrease in size in order to maintain the sameK.
Recall that K is also the SGS kinetic energy of the vortices lying in region (II) and should be obtained from the structure-function-matching procedure local to the vortex location. However, since, by construction, no grid points are placed within region (II), we will use the grid points just inside the LES domain, centered on the plane,z=h=h0+ ∆z, for this purpose.
In region (II), we now considerTxz as constant in (4.18) and model this as the geometric average of its value at the true wall and at the top of region (II), so that
−Txz =uτ
−Txz|e
Se
1/2 . Hence (4.18) becomes
K1= γIIK1/2 2
−Txz|e
Se
1/2. (4.19)
Equation (4.19), used in the limit κc → 0 for increased robustness (to avoid division by small numbers), provides a way to calculate the local K´arm´an constant. Solving (4.17) in region (II), hν< z≤h0, and evaluating the result atz=h0yields
eu|h0 = uτ K1
log h0
hν
+eu|hν =uτ 1
K1
log h0
hν
+h+ν
, (4.20)
where the constant of integration has been chosen by putting eu|hν = uτh+ν. We will use (4.20), which serves as a jump condition between the hν–h0 planes, along with ev = 0 andwe = 0, to set the Dirichlet boundary conditions at the lifted virtual wall h0; uτ is obtained from the solution of (4.15).
Equation (4.20) has been obtained from a physical model of region (II) in which the dominant Reynolds shear stress is modeled by streamwise-aligned vortices that transport low-momentum mean streamwise velocity away from the wall and high-momentum mean streamwise velocity toward the wall. The idea that these self-similar vortices scale only withz—independent ofl+ andδ—implies an overlap argument. Equation (4.20) couples eu|h0 with the resolved-scale LES in region (III), h0 < z ≤δ, which provides both K and e
Se. Equation (4.20) contains two constants, γ and h+ν. The latter is given empirically byh+ν = 11 from our discussion of region (I); a different value could be used for rough-wall flows. This physical model provides a means of dynamically calculating the instantaneous local “K´arm´an constant”K1 as part of the LES. This will be demonstrated later.
It appears from the present construction that we have confined attached eddies to region (II).
This is not the case; the existence of attached eddies in region (II) does not preclude their existence in region (III). We have only assumed that region (II) is dominated by attached eddies, which led us to explicitly model their dynamics. If they exist in region (III) and have sizes larger than the grid spacing,z >∆x, they would be directly captured by the LES. On the other hand, if they exist in region (III) and have sizes smaller than the grid spacing,z <∆x, they would be modeled by the SGS model.
One may also wonder if these attached eddies still exist near the top of region (II), h0, in very high Reynolds number flows since we designedh0to be fixed relative toδ. In the present simulations h+0 is as high as 149 k. Note, however, that the attached-eddy idea–that they have sizez and have characteristic velocityuτ–is intimately linked with the overlap argument that says that bothz and uτare the relevant parameters in the overlap region,ν/uτ zδ. In other words, as long ash0/δ is small, it does not matter how largeh+0 is for bothz anduτ to emerge as the relevant parameters, the necessary ingredients for attached eddies.
4.3.5 Estimation of the Mixing Time Constant γ
IIA constant, γII, is required in (4.20). Owing to the highly anisotropic character of near-wall tur- bulent physics, this is expected to be somewhat different in value from that used in the SGS scalar application (Pullin 2000).
Consider the interface of regions (II) and (III),z=h0, where both inner and outer layer modeling ideas are valid; in the spirit of LES filtering, we interpret this interface as a blurred boundary between the two regions so that the change in underlying vortical flow is gradual. This interface, z≈h0<0.1δ, is near the wall, so the LES filtered flow field can be approximated by simple shear
flow,∂uei/∂xj =δi1δj3S, whereS= deu/dz. This implies that, in this region, e
Se= (1/√
2,0,1/√ 2);
that is the detached vortices are inclined at 45◦to the wall.
We estimateγIIby matching Townsend’s structure parameter, a1=T13/Tii=T13/(2K),
of the two vortical flow descriptions at this interface region. Given the vortex alignment, this parameter measures the amount of shear stress that can be supported relative to the vortex kinetic energy. First, thee
Se alignment and (4.5) give a1|e
Se = (−K|e
Se/2)/(2K|e
Se) =−1/4.
Similarly, theex alignment and (4.5) give
a1|ex= (−γII(K|ex)1/2∆cS/2)/(2K|ex) =−γII∆cS(K|ex)−1/2/4.
To proceed, we assume highRe so that κc →0. Also, for simplicity, d→0. The subgrid kinetic energy of the streamwise vortex then reduces to
K|ex = 2h(δui)2i/ π2hd2i .
The local averaging is dominated by the background shear, so we can approximate h(δuei)2i ≈ h(δeu)2i ≈(∆zS)2, and for the same reasonhd2i= (∆z/∆c)2. Then
K|ex= 2 (∆cS/π)2.
Using this, we obtaina1for theex aligned vortex,
a1|ex=−2−5/2π γII. Finally, matching these,a1|e
Se =a1|ex, we haveγII= 21/2/π≈0.45. This is the value used presently for all LES.
4.3.6 Summary of SGS Wall Model
Our SGS model for the near-wall dynamics can be summarized as follows: for every cell adjacent to both the top and bottom walls, (4.15) is solved for η0 with terms on the right-hand side provided by the LES at the top of the wall-adjacent cell at z =h= ∆z+h0. This providesη0(x, y, t) and thusuτ(x, y, t). Equation (4.20) is then used to evaluate the streamwise slip velocityeu|h0(x, y, t) at
z=h0, withK1evaluated from (4.19) and withKandTxz|e
Se
evaluated atz=h= ∆z+h0from the LES structure-function-matching procedure. The other boundary conditions at the virtual wall are taken asev|h0(x, y, t) =w|eh0(x, y, t) = 0. This method couples the LES with the modeled, near-wall dynamics. The LES has implicit knowledge of the true no-slip boundary condition, because this was used in obtaining (4.15), and the smooth-wall condition through use ofh+ν = 11. Because the LES quantities in both (4.15) and (4.19) are evaluated at the top of the first cell atz =h= ∆z+h0, the height of the virtual wall atz=h0should satisfy hν < h0< h. Presently we useh0= 0.18 ∆z, independent of the LES resolution, and consider this as part of the overall grid. Some tests to investigate sensitivity toh0 were performed.